Complete Asymptotics for Solution of Singularly Perturbed Dynamical Systems with Single Well Potential

Abstract

We consider a singularly perturbed boundary value problem(-epsilon 2 increment + backward difference V center dot backward difference )u epsilon=0in omega,u epsilon=fon partial differential omega,f is an element of C infinity( partial differential omega).The functionVis supposed to be sufficiently smooth and to have the only minimum in the domain omega. This minimum can degenerate. The potentialVhas no other stationary points in omega and its normal derivative at the boundary is non-zero. Such a problem arises in studying Brownian motion governed by overdamped Langevin dynamics in the presence of a single attracting point. It describes the distribution of the points at the boundary partial differential omega, at which the trajectories of the Brownian particle hit the boundary for the first time. Our main result is a complete asymptotic expansion foru epsilon as epsilon ->+0. This asymptotic is a sum of a termK epsilon psi epsilon and a boundary layer, where psi epsilon is the eigenfunction associated with the lowest eigenvalue of the considered problem andK epsilon is some constant. We provide complete asymptotic expansions for bothK epsilon and psi epsilon; the boundary layer is also an infinite asymptotic series power in epsilon. The error term in the asymptotics foru epsilon is estimated in various norms.